Problem: Factor the following expression: $7$ $x^2$ $-12$ $x$ $-4$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(-4)} &=& -28 \\ {a} + {b} &=& & & {-12} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-28$ and add them together. Remember, since $-28$ is negative, one of the factors must be negative. The factors that add up to ${-12}$ will be your ${a}$ and ${b}$ When ${a}$ is ${2}$ and ${b}$ is ${-14}$ $ \begin{eqnarray} {ab} &=& ({2})({-14}) &=& -28 \\ {a} + {b} &=& {2} + {-14} &=& -12 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {7}x^2 +{2}x {-14}x {-4} $ Group the terms so that there is a common factor in each group: $ ({7}x^2 +{2}x) + ({-14}x {-4}) $ Factor out the common factors: $ x(7x + 2) - 2(7x + 2) $ Notice how $(7x + 2)$ has become a common factor. Factor this out to find the answer. $(7x + 2)(x - 2)$